National Academy Science Letters

, Volume 41, Issue 4, pp 243–247 | Cite as

Some Results on Tensor Product of a Graph and an Odd Cycle

  • Deepa Sinha
  • Pravin Garg
Short Communication


The tensor product \(G \times H\) of two graphs G and H is a graph such that the vertex set of \(G \times H\) is the cartesian product \(V(G) \times V(H)\) and two vertices \((u_1, u_2)\) and \((v_1, v_2)\) are adjacent in \(G \times H\) if and only if \(u_1\) is adjacent to \(v_1\) in G and \(u_2\) is adjacent to \(v_2\) in H. In this paper, we establish a structural characterization of \(G \times C_{2n+1}\). Further, we discuss Eulerian, Hamiltonian and planar properties of \(G \times C_{2n+1}\).


Tensor product of graphs Decomposable graphs Eulerian graphs Hamiltonian graphs Planar graphs 

Mathematics Subject Classification

05C75 05C76 



The authors express their gratitude to Prof. E. Sampathkumar who in his early research has brought up the idea of tensor product of the particular graphs which on reading gave us an instant insight to go for \(G \times C_{2n+1}\) and admire the beauty of traversability and many interesting properties of the structure.


  1. 1.
    Sabidussi G (1960) Graph multiplication. Math Z 72:446–457MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Vizing VG (1963) The cartesian product of graphs. Vyc Sis 9:30–43MathSciNetGoogle Scholar
  3. 3.
    Imrich W, Klavžar S (2000) Product graphs: structure and recognition. Wiley, New YorkMATHGoogle Scholar
  4. 4.
    Ĉulík K (1958) Zur theorie der graphen. Ĉasopis pro Pêstování Matematiky 83(2):133–155MathSciNetMATHGoogle Scholar
  5. 5.
    Weichsel PM (1962) The kronecker product of graphs. Proc Am Math Soc 13:47–52MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bosak J (1991) Decompositions of graphs. Kluwer Academic Publication, DordrechtMATHGoogle Scholar
  7. 7.
    Miller DJ (1968) The categorical product of graphs. Can J Math 20:1511–1521MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bermond JC (1978) Hamiltonian decompositions of graphs, digraphs, hypergraphs. Ann Discrete Math 3:21–28MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Capobianco MF (1970) On characterizing tensor-composite graphs. Ann N Y Acad Sci 175:80–84ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Harary F, Wilcox G (1967) Boolean operations on graphs. Math Scand 20:41–51MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Asmerom GA (1998) Imbeddings of the tensor product of graphs where the second factor is a complete graph. Discrete Math 182:13–19MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bottreau A, Métivier Y (1998) Some remarks on the Kronecker product of graphs. Inf Proc Lett 68(2):55–61MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Farzan M, Waller DA (1977) Kronecker products and local joins of graphs. Can J Math 29:255–269MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Harary F, Trauth CA Jr (1966) Connectedness of products of two directed graphs. SIAM J Appl Math 14(2):250–254MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Acharya BD (1975) Contributions to the theories of hyper grapahs, graphoids and graphs, Ph.D. Thesis, Indian Institute of Technology, BombayGoogle Scholar
  16. 16.
    Sampathkumar E (1975) On tensor product graphs. J Aust Math Soc 20(A):268–273MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Harary F (1969) Graph theory. Addison-Wesley Publishing Company, ReadingCrossRefMATHGoogle Scholar
  18. 18.
    West DB (1996) Introduction to graph theory. Prentice-Hall of India Pvt. Ltd, DelhiMATHGoogle Scholar
  19. 19.
    Gravier S (1997) Hamiltonicity of the cross product of two Hamiltonian graphs. Discrete Math 170:253–257MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Dirac GA (1952) Some theorems on abstract graphs. Proc Lond Math Soc 2(3):69–81MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Acharya M, Sinha D (2006) Common-edge sigraphs. AKCE Int J Graphs Comb 3(2):115–130MathSciNetMATHGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2018

Authors and Affiliations

  1. 1.Department of MathematicsSouth Asian UniversityChanakyapuri, New DelhiIndia
  2. 2.University of RajasthanJaipurIndia

Personalised recommendations